Eigenvalue Optimisation on Flat Tori and Lattice Points in Anisotropically Expanding Domains

Lagacé, Jean

Canadian journal of mathematics, Tome 72 (2020) no. 4, pp. 967-987

Résumé

This paper is concerned with the maximisation of the $k$-th eigenvalue of the Laplacian amongst flat tori of unit volume in dimension $d$ as $k$ goes to infinity. We show that in any dimension maximisers exist for any given $k$, but that any sequence of maximisers degenerates as $k$ goes to infinity when the dimension is at most 10. Furthermore, we obtain specific upper and lower bounds for the injectivity radius of any sequence of maximisers. We also prove that flat Klein bottles maximising the $k$-th eigenvalue of the Laplacian exhibit the same behaviour. These results contrast with those obtained recently by Gittins and Larson, stating that sequences of optimal cuboids for either Dirichlet or Neumann boundary conditions converge to the cube no matter the dimension. We obtain these results via Weyl asymptotics with explicit control of the remainder in terms of the injectivity radius. We reduce the problem at hand to counting lattice points inside anisotropically expanding domains, where we generalise methods of Yu. Kordyukov and A. Yakovlev by considering domains that expand at different rates in various directions.

Détail

DOI : 10.4153/S0008414X19000130

Mots-clés : spectral optimisation, Laplacian, eigenvalue, asymptotics

Bibliographie

[1] Antunes, P. R. S. and Freitas, P., Numerical optimization of low eigenvalues of the Dirichlet and Neumann Laplacians. J. Optim. Theory Appl. 154(2012), 235–257. https://doi.org/10.1007/s10957-011-9983-3 Google Scholar

[2] Antunes, P. R. S. and Freitas, P., Optimal spectral rectangles and lattice ellipses. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469(2013), no. 2015, 20120492. https://doi.org/10.1098/rspa.2012.0492 Google Scholar

[3] Banaszczyk, W., New bounds in some transference theorems in the geometry of numbers. Math. Ann. 296(1993), 625–635. https://doi.org/10.1007/BF01445125 Google Scholar

[4] Van Den Berg, M., Bucur, D., and Gittins, K., Maximising Neumann eigenvalues on rectangles. Bull. Lond. Math. Soc. 48(2016), 877–894. https://doi.org/10.1112/blms/bdw049 Google Scholar

[5] Van Den Berg, M. and Gittins, K., Minimizing Dirichlet eigenvalues on cuboids of unit measure. Mathematika 63(2017), 469–482. https://doi.org/10.1112/S0025579316000413 Google Scholar

[6] Berger, A., The eigenvalues of the Laplacian with Dirichlet boundary condition in ℝ2 are almost never minimized by disks. Ann. Global Anal. Geom. 47(2015), 285–304. https://doi.org/10.1007/s10455-014-9446-9 Google Scholar

[7] Berger, M., Gauduchon, P., and Mazet, E., Le spectre d’une variété riemannienne. Lecture Notes in Mathematics, 194, Springer-Verlag, Berlin-New York, 1971. Google Scholar

[8] Buser, P., A note on the isoperimetric constant. Ann. Sci. École Norm. Sup. (4) 15(1982), 213–230. Google Scholar

[9] Cassels, J. W. S., An introduction to the geometry of numbers. Die Grundlehren der mathematischen Wissenschaften, Band 99, Springer-Verlag, Berlin-New York, 1971. Google Scholar

[10] Colbois, B. and Dodziuk, J., Riemannian metrics with large 𝜆. Proc. Amer. Math. Soc. 122(1994), 905–906. https://doi.org/10.2307/2160770 Google Scholar

[11] Duistermaat, J. J. and Guillemin, V. W., The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29(1975), 39–79. https://doi.org/10.1007/BF01405172 Google Scholar

[12] Faber, G., Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt. Sitzungsber. Bayer. Akad. Wiss. München, Math.-Phys. Kl., 1923, pp. 169–172. Google Scholar

[13] Gittins, K. and Larson, S., Asymptotic behaviour of cuboids optimising Laplacian eigenvalues. Integral Equations Operator Theory 89(2017), 607–629. https://doi.org/10.1007/s00020-017-2407-5 Google Scholar

[14] Hassannezhad, A., Kokarev, G., and Polterovich, I., Eigenvalue inequalities on Riemannian manifolds with a lower Ricci curvature bound. J. Spectr. Theory 6(2016), 807–835. https://doi.org/10.4171/JST/143 Google Scholar

[15] Hersch, J., Quatre propriétés isopérimétriques de membranes sphériques homogènes. C. R. Acad. Sci. Paris Sér. A-B 270(1970), A1645–A1648. Google Scholar

[16] Iosevich, A. and Liflyand, E., Decay of the Fourier transform, analytic and geometric aspects. Birkhäuser/Springer, Basel, 2014. https://doi.org/10.1007/978-3-0348-0625-1 Google Scholar

[17] Kao, C.-Y., Lai, R., and Osting, B., Maximization of Laplace-Beltrami eigenvalues on closed Riemannian surfaces. ESAIM Control Optim. Calc. Var. 23(2017), 685–720. https://doi.org/10.1051/cocv/2016008 Google Scholar

[18] Karpukhin, M., Nadirashvili, N., Penskoi, A. V., and Polterovich, I., An isoperimetric inequality for Laplace eigenvalues on the sphere. J. Diff. Geom., to appear. Google Scholar

[19] Kordyukov, Yu. A. and Yakovlev, A. A., Lattice points in domains and adiabatic limits. (Russian) Algebra i Analiz 23(2011), 80–95. https://doi.org/10.1090/S1061-0022-2012-01225-2 Google Scholar

[20] Kordyukov, Yu. A. and Yakovlev, A. A., The problem of the number of integer points in families of anisotropically expanding domains, with applications to spectral theory. Mat. Zametki 92(2012); trans. in Math. Notes 92(2012), no. 3–4, 574–576. https://doi.org/10.1134/S0001434612090295 Google Scholar

[21] Kordyukov, Yu. A. and Yakovlev, A. A., The number of integer points in a family of anisotropically expanding domains. Monatsh. Math. 178(2015), 97–111. https://doi.org/10.1007/s00605-015-0787-7 Google Scholar

[22] Kordyukov, Yu. A. and Yakovlev, A. A., On a problem in geometry of numbers arising in spectral theory. Russ. J. Math. Phys. 22(2015), 473–482. https://doi.org/10.1134/S106192081504007X Google Scholar

[23] Krahn, E., Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises. Math. Ann. 94(1925), 97–100. https://doi.org/10.1007/BF01208645 Google Scholar

[24] Krahn, E., Über Minimaleigenschaften der Kugel in drei und mehr Dimensionen. Acta Comm. Univ. Tartu (Dorpat) A9(1926), 1–44. Google Scholar

[25] Lagacé, J. and Parnovski, L., A generalised Gauss circle problem and integrated density of states. J. Spectr. Theory 6(2016), 859–879. https://doi.org/10.4171/JST/145 Google Scholar

[26] Nadirashvili, N., Berger’s isoperimetric problem and minimal immersions of surfaces. Geom. Funct. Anal. 6(1996), 877–897. https://doi.org/10.1007/BF02246788 Google Scholar

[27] Szegö, G., Inequalities for certain eigenvalues of a membrane of given area. J. Rational Mech. Anal. 3(1954), 343–356. https://doi.org/10.1512/iumj.1954.3.53017 Google Scholar

[28] Weinberger, H. F., An isoperimetric inequality for the N-dimensional free membrane problem. J. Rational Mech. Anal. 5(1956), 633–636. https://doi.org/10.1512/iumj.1956.5.55021 Google Scholar